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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimpnfxnegmnf2 | Structured version Visualization version GIF version |
Description: A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
xlimpnfxnegmnf2.j | ⊢ Ⅎ𝑗𝐹 |
xlimpnfxnegmnf2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimpnfxnegmnf2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimpnfxnegmnf2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
xlimpnfxnegmnf2 | ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimpnfxnegmnf2.j | . . 3 ⊢ Ⅎ𝑗𝐹 | |
2 | xlimpnfxnegmnf2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | xlimpnfxnegmnf2.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 1, 2, 3 | xlimpnfxnegmnf 43846 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
5 | xlimpnfxnegmnf2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | 1, 5, 2, 3 | xlimpnf 43874 | . 2 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
7 | nfmpt1 5212 | . . . 4 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)) | |
8 | 3 | ffvelcdmda 7030 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*) |
9 | 8 | xnegcld 13149 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝑒(𝐹‘𝑘) ∈ ℝ*) |
10 | nfcv 2906 | . . . . . 6 ⊢ Ⅎ𝑘-𝑒(𝐹‘𝑗) | |
11 | nfcv 2906 | . . . . . . . 8 ⊢ Ⅎ𝑗𝑘 | |
12 | 1, 11 | nffv 6848 | . . . . . . 7 ⊢ Ⅎ𝑗(𝐹‘𝑘) |
13 | 12 | nfxneg 43491 | . . . . . 6 ⊢ Ⅎ𝑗-𝑒(𝐹‘𝑘) |
14 | fveq2 6838 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
15 | 14 | xnegeqd 43467 | . . . . . 6 ⊢ (𝑗 = 𝑘 → -𝑒(𝐹‘𝑗) = -𝑒(𝐹‘𝑘)) |
16 | 10, 13, 15 | cbvmpt 5215 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)) = (𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)) |
17 | 9, 16 | fmptd 7057 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)):𝑍⟶ℝ*) |
18 | 7, 5, 2, 17 | xlimmnf 43873 | . . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥)) |
19 | 2 | uztrn2 12716 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍) |
20 | xnegex 13057 | . . . . . . . . 9 ⊢ -𝑒(𝐹‘𝑗) ∈ V | |
21 | fvmpt4 43259 | . . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ -𝑒(𝐹‘𝑗) ∈ V) → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) = -𝑒(𝐹‘𝑗)) | |
22 | 20, 21 | mpan2 690 | . . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) = -𝑒(𝐹‘𝑗)) |
23 | 22 | breq1d 5114 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ -𝑒(𝐹‘𝑗) ≤ 𝑥)) |
24 | 19, 23 | syl 17 | . . . . . 6 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ -𝑒(𝐹‘𝑗) ≤ 𝑥)) |
25 | 24 | ralbidva 3171 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → (∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
26 | 25 | rexbiia 3094 | . . . 4 ⊢ (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥) |
27 | 26 | ralbii 3095 | . . 3 ⊢ (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥) |
28 | 18, 27 | bitrdi 287 | . 2 ⊢ (𝜑 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
29 | 4, 6, 28 | 3bitr4d 311 | 1 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2886 ∀wral 3063 ∃wrex 3072 Vcvv 3444 class class class wbr 5104 ↦ cmpt 5187 ⟶wf 6488 ‘cfv 6492 ℝcr 10984 +∞cpnf 11120 -∞cmnf 11121 ℝ*cxr 11122 ≤ cle 11124 ℤcz 12433 ℤ≥cuz 12697 -𝑒cxne 12960 ~~>*clsxlim 43850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-1o 8380 df-er 8582 df-pm 8702 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fi 9281 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-z 12434 df-uz 12698 df-xneg 12963 df-ioo 13198 df-ioc 13199 df-ico 13200 df-icc 13201 df-topgen 17261 df-ordt 17319 df-ps 18391 df-tsr 18392 df-top 22171 df-topon 22188 df-bases 22224 df-lm 22508 df-xlim 43851 |
This theorem is referenced by: (None) |
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